Theorem lgsdinn0 15938

Description: Variation on lgsdi 15927 valid for all M , N but only for positive A. (The exact location of the failure of this law is for A = -u 1, M = 0, and some N in which case ( -u 1 /L 0 ) = 1 but ( -u 1 /L N ) = -u 1 when -u 1 is not a quadratic residue mod N.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Assertion

Ref	Expression
lgsdinn0	|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Proof of Theorem lgsdinn0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6060	. . . . . . . . 9 |- ( x = N -> ( A /L x ) = ( A /L N ) )
2	1	oveq1d 6067	. . . . . . . 8 |- ( x = N -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
3	2	eqeq2d 2246	. . . . . . 7 |- ( x = N -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) )
4		sq1 10999	. . . . . . . . . . . . . . . . 17 |- ( 1 ^ 2 ) = 1
5	4	eqeq2i 2245	. . . . . . . . . . . . . . . 16 |- ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> ( A ^ 2 ) = 1 )
6		nn0re 9507	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN0 -> A e. RR )
7		nn0ge0 9523	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN0 -> 0 <_ A )
8		1re 8275	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
9		0le1 8757	. . . . . . . . . . . . . . . . . . 19 |- 0 <_ 1
10		sq11 10978	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
11	8, 9, 10	mpanr12 439	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
12	6, 7, 11	syl2anc 411	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
13	12	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
14	5, 13	bitr3id 194	. . . . . . . . . . . . . . 15 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 <-> A = 1 ) )
15	14	biimpa 296	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A = 1 )
16	15	oveq1d 6067	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = ( 1 /L x ) )
17		1lgs 15933	. . . . . . . . . . . . . 14 |- ( x e. ZZ -> ( 1 /L x ) = 1 )
18	17	ad2antlr 489	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 /L x ) = 1 )
19	16, 18	eqtrd 2267	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = 1 )
20	19	oveq1d 6067	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( 1 x. ( A /L 0 ) ) )
21		nn0z 9599	. . . . . . . . . . . . . . 15 |- ( A e. NN0 -> A e. ZZ )
22	21	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A e. ZZ )
23		0z 9590	. . . . . . . . . . . . . 14 |- 0 e. ZZ
24		lgscl 15904	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A /L 0 ) e. ZZ )
25	22, 23, 24	sylancl 413	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. ZZ )
26	25	zcnd 9704	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. CC )
27	26	mullidd 8294	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 x. ( A /L 0 ) ) = ( A /L 0 ) )
28	20, 27	eqtr2d 2268	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
29		lgscl 15904	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
30	21, 29	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
31	30	zcnd 9704	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. CC )
32	31	adantr 276	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L x ) e. CC )
33	32	mul01d 8668	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. 0 ) = 0 )
34	21	adantr 276	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> A e. ZZ )
35		lgs0 15903	. . . . . . . . . . . . . 14 |- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
36	34, 35	syl 14	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
37		ifnefalse 3635	. . . . . . . . . . . . 13 |- ( ( A ^ 2 ) =/= 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
38	36, 37	sylan9eq 2287	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = 0 )
39	38	oveq2d 6068	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L x ) x. 0 ) )
40	33, 39, 38	3eqtr4rd 2278	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
41		zsqcl 10976	. . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
42	34, 41	syl 14	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A ^ 2 ) e. ZZ )
43		1z 9605	. . . . . . . . . . . 12 |- 1 e. ZZ
44		zdceq 9655	. . . . . . . . . . . 12 |- ( ( ( A ^ 2 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
45	42, 43, 44	sylancl 413	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ x e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
46		dcne 2425	. . . . . . . . . . 11 |- (DECID ( A ^ 2 ) = 1 <-> ( ( A ^ 2 ) = 1 \/ ( A ^ 2 ) =/= 1 ) )
47	45, 46	sylib 122	. . . . . . . . . 10 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 \/ ( A ^ 2 ) =/= 1 ) )
48	28, 40, 47	mpjaodan 806	. . . . . . . . 9 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
49	48	ralrimiva 2617	. . . . . . . 8 |- ( A e. NN0 -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
50	49	3ad2ant1 1045	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
51		simp3 1026	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
52	3, 50, 51	rspcdva 2928	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
53	52	adantr 276	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
54	21	3ad2ant1 1045	. . . . . . . . 9 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A e. ZZ )
55	54, 23, 24	sylancl 413	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. ZZ )
56	55	zcnd 9704	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. CC )
57	56	adantr 276	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) e. CC )
58		lgscl 15904	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
59	54, 51, 58	syl2anc 411	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
60	59	zcnd 9704	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. CC )
61	60	adantr 276	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L N ) e. CC )
62	57, 61	mulcomd 8297	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L 0 ) x. ( A /L N ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
63	53, 62	eqtr4d 2270	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L 0 ) x. ( A /L N ) ) )
64		oveq1 6059	. . . . . 6 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
65	51	zcnd 9704	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
66	65	mul02d 8667	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
67	64, 66	sylan9eqr 2289	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = 0 )
68	67	oveq2d 6068	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
69		simpr 110	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 )
70	69	oveq2d 6068	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L M ) = ( A /L 0 ) )
71	70	oveq1d 6067	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L 0 ) x. ( A /L N ) ) )
72	63, 68, 71	3eqtr4d 2277	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
73		oveq2 6060	. . . . . . . 8 |- ( x = M -> ( A /L x ) = ( A /L M ) )
74	73	oveq1d 6067	. . . . . . 7 |- ( x = M -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
75	74	eqeq2d 2246	. . . . . 6 |- ( x = M -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) )
76		simp2 1025	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
77	75, 50, 76	rspcdva 2928	. . . . 5 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
78	77	adantr 276	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
79		oveq2 6060	. . . . . 6 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
80	76	zcnd 9704	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
81	80	mul01d 8668	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
82	79, 81	sylan9eqr 2289	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = 0 )
83	82	oveq2d 6068	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
84		simpr 110	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> N = 0 )
85	84	oveq2d 6068	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L N ) = ( A /L 0 ) )
86	85	oveq2d 6068	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
87	78, 83, 86	3eqtr4d 2277	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
88	72, 87	jaodan 805	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
89		neanior 2501	. . 3 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
90		lgsdi 15927	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
91	21, 90	syl3anl1 1322	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
92	89, 91	sylan2br 288	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
93		zdceq 9655	. . . . 5 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
94	76, 23, 93	sylancl 413	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID M = 0 )
95		zdceq 9655	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
96	51, 23, 95	sylancl 413	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID N = 0 )
97		dcor 944	. . . 4 |- (DECID M = 0 -> (DECID N = 0 -> DECID ( M = 0 \/ N = 0 ) ) )
98	94, 96, 97	sylc 62	. . 3 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
99		exmiddc 844	. . 3 |- (DECID ( M = 0 \/ N = 0 ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
100	98, 99	syl 14	. 2 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
101	88, 92, 100	mpjaodan 806	1 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   ifcif 3622   class class class wbr 4111  (class class class)co 6052   CCcc 8127   RRcr 8128   0cc0 8129   1c1 8130   x. cmul 8134   <_ cle 8311   2c2 9290   NN0cn0 9498   ZZcz 9579   ^cexp 10904   /Lclgs 15887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-sup 7277  df-inf 7278  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-fl 10634  df-mod 10689  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-proddc 12241  df-dvds 12478  df-gcd 12654  df-prm 12809  df-phi 12912  df-pc 12987  df-lgs 15888

This theorem is referenced by: (None)